Scale Models The simple approximations used to explore a portfolio's behaviour over various time horizons hold traps for the

unwary. Francis Diebold, Andrew Hickman, Atsushi lnoue and Til Schuermann explain

' hat is the relevant horizon for risk m;lnagement? This obvious question has no obvious answer. ' I-Iorizons sucli as seven to 10 d;lys for equity ancl foreign exchange ancl 30 clays for interest rate in- struments are routinely cliscussecl. 111 Ihct, even horizons ns long as ;I yes]. are not ~incomliion.'

Ol~c~:~rionally, 1151i is often assesseel at a one-clay horizon, anel even sliort- er (i~ir~.:~-clay) horizons have I~een cliscussecl. Short-horizon risk measures :Ire commolily converteel to other horizons, sucli :IS 10-clay or 30-cl;1y, Iby scnling.' For esaliilble, to obtain :I 10-day volatility we \wo~~ltl multiply ~ l l e one-clay vol:~tiliry Iby f i . Mol.eover, the horizon conversion is often sig- nific:~ntly longer than 10 clays. Many banks, for example, link the men- su1,ement of rl.acling volatility to internal capital allocation nntl risk-aclji~stecl perfornlance measurement schemes, which rely on annual volatility esti- ~i i ;~tes. Tlie tempt;ltion is to s a l e one-clay vol;ltility by &

The roiltilie anel uncritical else of sc;lling is also wiclely accelltecl 11y reg- ~11:1tors. For example, it is a prominent feature of the Basle Commitlee o n Banlting Supervision's Januzlry 1996 A1ner7d1iie1tl lo /he C~ lp i l a l Acco~.cl lo I17co1po1nte 1lll~1r6et Risks. Specifically, tlie ;~~nenclment requires :I 10-clay holeling lperiocl ancl ;~cLvises conversion Ily scaling: "In c:~lculating vnl~ie- ;I[-risk, ;III inst;~ntaneous price shoclc eq~~iva len t to a 10-clay movenienl in prices is to Ibe ~lsecl, ie, the ~ n i n i m ~ i ~ i i "l~olcling periocl" will Ibe 10 tl~cling cl~~ys. Banlis may use value-at-risk numbers calcul;~ted accorcling to short- e r liolcling ~~eriocls sc;~lecl LIII to 10 days by the sqwlre root of tiliie ..." (page 44, section B4, paragraph c).

In this paper we souncl an alarm: sucli scaling is inappropriate and mis- 1e;lcling. \Ve show Ibelow that converting volatilities by scaling is st:~tisti- cally appropriate only under strict conclitions that are routinely viol;~tecl by high-heqi~ency (eg, one-clay) asset returns. Even in the unlikely event tliat tlie conclitions for its statistical legitimacy :Ire met, sc;~ling is nevertlieless prol~lematic for economic reasons associareel with fluctuations in portfo- lio composition.

Short and long horizons

'~y-"\ ul. lbasic finclings can Ibe summ:~rised as: scaling worlts in inclepenclent, identically clistributed (iicl) environments bur Fails othelwise. Here w e describe tlie restrictive environ- ment in which scaling is appropriate. Let V, Ibe a log price

at t i ~ n e t ~ ~ n c l suppose that changes in tlie log price are independently ant1 iclentically distrib~ltecl:

iid Vt = V t - 1 + Et ; E t -(0,02)

where the one-clay return is E, with staliclarcl cleviation a. Similarly, the h- clay return is:

with variance ha2 and stanclarcl deviation AD. Hence tlie "rule": to con- vert a one-day standard deviation to an h-day standarcl cleviation, simply scale Iby dh. For some applications, a percentile of the distribution of h-day

returns may Ile clesiretl; percentiles also scale Iby dh if log changes are nor only iicl I I L I ~ also n o r ~ i ~ ; ~ l I y distrilxltecl

Tlie ~~rescrilbecl scnling rule relies o n one-clay returns being incleprn- clent ;~ncl iclentically clistl.ih~ltecl. The l i te~ltul-e o n mean reversion in stock returns appreciates this, nntl scaling is often ~isecl ;IS :I test for whether returns are iicl, ranging kom erlrly work (eg, Cootner, 1964) to recenl work (eg, Camplbell, Lo ancl Macl<i~ilay, 1997). But high-freq~~ency fi- nancial asset returns are clistinctly 1701 inclepenclelic r~ncl iclenric:llly clis- trilbiltecl. Even if high-freqi~ency portfolio returns are conclition;~l-mean inclepenclent (which has I>eeli tlie sul~jec[ of intense elelbare in tlie effi- cient m;trltets liter;lt~~re), thcy are cer t~ in ly not colitli~io~i;~I-v;~rii~nce in- depenclent, as shown in li~lnclrecls of' recent p:lpers tlocuinenting strong volatility persistence in financial ;Isset returns. '

To higliliglit the failure of sc:~ling in non-iicl environliients ancl the nil- lure of the associatecl erroneous long-horizon volnriliry esrim:~res, \ye \vill use a simple GarcIi(1,l) process for one-clay returns:

where t = 1, ..., T. \Ve impose the i~suz~l regillal.it)/ ancl coval.iance sta- tionarity conclitions, 0 < o < -, a 2 0, p 2 0 ancl cl+P < 1. Tllc key fe'ea- ture of the G a r c l ~ ( l , l ) process is that it allows for time-valying conclitiollal \lolatility, which o c c ~ ~ r s when a ;lncl/or P is noli-zero. The ~noclel lias I ~ e e n tilted to hi~nclrecls of finz~ncial series ~ ~ n c l lias Ibeen tre~nencloi~sly .s~~ccessfill eml~irically; hence its pop~ilarity. ' \Ve Ilasten to aclcl, Iiowev- el; hat our general thesis - that scaling Fails in the non-iicl environments associateel wicli high-frequency asset returns - cloes not clel~end on any way on a Garch(l.1) structure. Rather, w e focus o n the Garch(1, l) case I~ecause i t lias Ibeen stuclied most intensively, yielcling n we:~lth of ~,esults that enal>le 11s to illustrate the fail~lre of scal i~ig I>oth analyt i~l l ly :~ncl 11y simulation.

Drost B Nijman (1993) stncly the temporal aggregation of Garch processes." Suppose w e I~egin with a sample path of 3 one-clay return series, {Y(~,,}: = which follows tlie Garch(1,l) process al~ove. ' Then Drost & Nijman s h o w that, under reg~ilarity conclitions, the corl.esponcl- ing sample pat11 of h-clay returns, {y(,,,,}:'l s i i i ~ i a ~ ~ l y follo.ivs ;I G;~rch( l , l ) process witli:

- -. - .-

' Chew (1994) provides insightful early discussion " leading example is Bankers Trust's Raroc system; see Falloon (7995) See, for example, Smithson & Minton (1996a, b) and JP Morgan (1996) See, for example, the surveys by Bollerslev, Chou & Kroner (1992) and Diebold &

Lopez (1 995) SAgain, see the surveys of Garch models in finance by Bollersiev. Chou & Kroner (1992) and Diebold & Lopez (1995)

More precisely, they define and study the temporal aggregation of weak Garch processes, a formal definition of which is beyond the scope of th~s paper. Although the distinction is not crucial for ourpurposes, technically inclined readers should read "weak Garch" whenever they encounter the word "Garch" 'Note the new and more cumbersome, but necessary, notation, the subscript in which keeps track of the aggregation level

104 RISK JANUARY 1998

I. One-day returns and volatilities

30 1 Observed series

-30 1 I I I I

0 2,000 4,000 6,000 8,000 Time

0 2,000 4,000 6,000 8,000 Time

and I P(,,, I <1 is the solution of the quadratic equation:

where:

l-(~+a)'~ b = (a - pa(P + a)) -----

~-((]+a)'

and K is the Iturtosis of y,.TIie Drost-Nijman for~nula is neither pretty nor intuitive, but it is important, because it is the key to correct conver- sion of one-clay volatility to h-day volatility. It is painfully ol~vious, more- over, that the simple scaling formula does not look at all like the Drost-Nijman formula.

If, how eve^; tlie scaling forniula were an accurate approxi~nation to the Drost-Nijman formula, it would still be very useful because of its simplic- ity and intuitive appeal. Unfo~.tunately, this is not the case. As h + -, analy- sis of the Drost-Nijman formula reveals that a(,, -t 0 and P(,, 4 0, which is to say that temporal aggregation produces gradual disappea1,ance of volatility fl~~ctuations.~ Scaling, in contrast, magnifies volatility fluctuations.

A detailed example ,T et us examine the failure of scaling in a specific example. We pa-

l rarneterise the Garch(1,l) process ro be I-ealistic For daily returns I by setting a = 0.10 and b = 0.85, which are typical of the pa- 7 rameter values obtained for estimated Garch(1,l) processes. The

choice of w is arbitrary and amounts to a nornialisation, or choice of scale.

2. True and scaled 10-day volatilities

,$ 50 1 Scaled volatility, m = 10 .-

i

S o l I I I I

0 2,000 4,000 6,000 8,000 Time :: 50 ] True volatility, rn = 10

2 40 u

C

S o l I I I I I I I I 1 0 100 200 300 400 500 600 700 800 900

Time

We set w = 1, wllich implies that (lie unconclitional variance of the [process equ;~ls 20. \We also set:

disc;~rcl the first 1,000 realisations to allow the effects of tlie initial condi- tion to clissipate, ancl keep the following T = 9 ,000 realisations. 111 figure 1, we show the series of claily returns and the corresponcling series of one- clay conclition;~l stanclarcl cleviations, ot. The daily volatility fluctuatio~is are evident.

Now we examine the 10-clay volatility, corresponding to h = 10 . In fig- ure 2, we show 10-clay volatilities calculatecl in nvo cliffel.ent ways. We 01,- tain [lie first (incorrect) 10-day volatility by scaling the one-clay volatili~y, o,, I J ~ m. We obtain the seconcl (correct) 10-clay volatility by applying the Drost-Nijman for~uula. '~

It is clear that altlioi~gh scaling procluces volatilities that are correct on average, i t magnifies the volatility flucti~ations, whereas they should in fact I J ~ lessened. That is, scaling produces erroneous conclusions of large fluc- tuations in tlie conclition;ll variance of long-horizon returns, when in f:icr tlie opposite is true. Moreover, we cannot clailn that the scalecl volatility estimates are "conselvative" in xny sense; rathei; they are sometimes too high and sometilnes too low.

If scaling is inappropriate, then what is appropriate? Firstly, as we have shown, if the short-horizon return model is correctly specifiecl as a Garch(1,l) process, then long-horizon volatilities can be calculated using the Drost-Nij~nan formula. Second, if the short-horizon return model is cor- rectly specified but does not fall into the family of models covered I J ~ Drost and Nijman, then tlie Drost-Nijnian results do not apply, and there are no known analytic rnetllods for computing h-day volatilities from one-clay volatilities. If we had analytic for~nulas, we coilld apply them, but we don't. So, if h-clay volatilities are of interest, i t makes sense to use an h-day ~noclel.

Tliird, when the one-day return ~iioclel is not correctly specifiecl, things

Bollerslev (1986) shows that a necessary and sufficient condition for the existence of a finite fourfh moment, and hence a finite kurtosis, is 3a2+2ap+P2 < 1 " The Drost-Nijman result coheres with the result of Diebold (1988), who shows Ihat temporal aggregation produces returns that approach an unconditional normal distri- but~on, which implies that volatility fluctuations must vanish

lo We set o:,,, at its unconditional mean

105 RISK JANUARY 1998

are even trickier." For example, the I x s t approxi l i ia t ion t o 10-clay return volatility t lyn;~mics may I le very clifferenr f rom what one gets 11y apply- ing 'lie Drost-Nijman forrnul:~ to an (incorrect) estim;~tecl Garch(1, I) moclel for one-clay 1-eti1r1i vol;ltility clyn;~mics (ant1 o f course very clifferent :IS

wel l 1'1.om what one gets I3y scaling estiniates o f tlaily ret1il.n volarilities I)y a). This ag;iin s~lggcsrs that i f h-cl;ly volarilities ;Ire o f interest, i t m:lltes sense to use ; ~ n h-clay moclel.

Conclusion I T he relevant horizon may vzuy 11y asset cl:lss (eg, ecl~l i ty vel:sils

fisecl incorn?), incl~lstry (eg, Ixlnlt ing \Tersus ins~lr;lnce), posi- t ion i n t l ie f irm (eg, tl.;~cling eleslt versus chiel' 1'inanci:ll ol'licer), anel motiv:~rion (eg, private versus ~ ~ - g i ~ l ; ~ t o ~ y ) , :lncl thought milst

be y \ . c n to the relev:lnt I ior izon o n a n ; ~ l ~ l ) l i c : ~ t i o ~ i - l ) ~ ~ - ; ~ l ~ l i c : ~ t i o ~ i I~ i ls is . hloclelling volatility only ;I! one sliol-r liol,izon, l i ) l lo\\~ecl I)y sc:lling to con- vert to longer hol.izons, is li ltely to I)e in;~ppropri;lte anel ~nisle:lcling, he- c;iuse tempor:~l :lggreg:~tion s l i o i ~ l d recluce volatiliry I l~ lc tuar ion.~, \vhere:ls scaling :lmplifies thelii. '? Insteatl, a strong case can I le ~n;lcle 1;~. ~ l s i n g clil- ferent moclels :I[ clilferent

\Ve hasten to ;~clcl that i t is not our intent to conclemn scaling :ll\v:lys ant1 e\/er)lwhere. Scaling is c I i :~r l~ i ing ly simple, ancl i t is :~ l ) lxo lx i :~te 1111cler cert;~in contlitions. hioreover, even when rliose contlirions :II.C violatccl, sc;llilig I)I.OCILIC~S resolts t l i i ~ t :\re correct o n avel.age, ;IS \ve sho\vecl. I-lence sc:~ling has its place, ant1 its witlesp~~e:lcl use ;IS :I loo1 Ibr :lp13rosirn;1!e hori- ~ 0 1 1 conversion is ilncle~~.stantlal,le. But :IS our so l~his t ic :~t ion incre>lses, [ l ie Ila\vs w i th such "fi~,st-genel.a!io~i" rules o f t l i n m l ~ hecome more p~~onouncecl , :111cl [ l ie lpossil~ilities li)r ilnl,rovement Iwcome ;lpp;lrent. O u r inlelit is to st i i l i~i late such i lnprove~nent .

\S/e helieve th;~t rlie ~ l s e ol'clifl'erent moclels for cliffel.ent 1io1.izo1-r~ is ;in

iliil)orL:liit stel) ill the riglit clirection. But even \vith tI1:1l sol~his1ic:ltee~ sl1.at- egy, the nilgging ancl routinely neglecteel ~)roblern o l ~ ~ o r l f o l i o Iluctri:~tions, pinpointecl i n :I prescient article I ly i<~ip icc & O'D1,ien (1995). relnains, hlea- suring the volatility ol'tl;~ding l.esults clepencl.s not only o n lie vol;ltility 01.

positions are laicl o f f i n the marltet 01- herlgecl. Seconcl, tmclers m:ly I ) L I ~ 011 (or tali? off) short-term speculative posilions, o r acljust long-term propri- ernly t ~ l c l i n g str:ltegies. Fin;~lly, trading man;lgement may intervene to I-e- tluce ~Ios i t i io is i n response to aclverse mnrltet mo\fements.

\V l ia teve~ tlie cause or tl~rcr~1ation.s i n the posit ion vector, i t conllicts \vitli tl ie h-clay I~ i~y - :~n t l -ho lc l :~ssu~i ip t ion. The clegree to which this as- s ~ t ~ i i l ~ t i o n is vioI:~tecI w i l l clepentl on the tfilcling clesk's I~usiness str:ltegy, the ins t r~ tme~ i ts i t tracles ant1 the liquiclity of the m:1rltet.s i n wh ich i t trncles. For es;~mple, even one clay may be too long :I horizon over \vl i ic l i l o :IS- sumc :I const;lnt portfol io I'ol a marltet maker i n ;I major European cur- rency - the elicl-of-cl:ty portfol io w i l l I>e;lr litrle I-el:ttion to l l ie v:~riety o f posi t io~is t l i ; ~ ~ c o ~ l l t l he t ~ i k e n over the course o f tlie nest cl;ly. ~ n ~ l c h less 11ie nest 10 clays. To ~inclerstantl the risk over :I longer horizon, \ve neccl nut only ro l>~lst st:~tislic:~l muclcls (or the ~ l n c l e r l y i ~ l g ~ i i ;~~ . l te l price vol:ltili- ty I x i t :lIso I~OIXIS~ l ~ e I i : l v i o ~ ~ ~ ~ l ~ i io t le l s for cI1;111ges i ~ i tc lc l i~ ig posit io~is.

Fin;llly, w e stress tlie chnllenges associ:~recl will1 :~gg~.eg:lring ~rislis:lci.oss p ~ s i t i o n s :II~CI tr:lcling clesks when he rislts :Ire assesseel :it clil.fe~.en~ l ior i- z o ~ l s . Ol)vio~lsly, one c l n n o l s imply atlcl togelher risk Ineasurcs :II clil'le1.- etlt horizons. Insteatl, conversion 10 ;I coln111on horizon intlsl I lc clolle t l i rougl i a coml3ination o f st;ltistic:llly :~plxol)ri;lte h-cl:ly ~noclels 01' pl.icc \lol;ltility :111d I~eh ;~ \~ iou~ .a l motlels i o r ch;~nge.s i n 1~1tlers' ~~os i t i ons . ' f l l ; ~ ~ , i n o ~ i r view, is :I pressing clirection I'or Tuu~re reseal.cli. W

Francis Diebold is professor of economics and statistics at the Uni- versity of Pennsylvania and a member of the Oliver, Wyman Institute. Andrew Hickman, formerly of Oliver. Wyman & Com~anv, is an expert in quantitative risk management. ~ t ' s u i h i lnoue is a'phD'candidate in economics at the University of Pennsylvania. Til Schuermann is head of research at Oliver, ~ y m a n & company. The first auihor would like to thank the Oliver, Wyman Institute for its hospitality and inspiration. Peter Christoffersen, Paul Kupiec, Jose Lopez and Tony Santomero provided helpful discussion, as did participants at the Federal Re- serve Bank of Atlanta Conference on Financial Market structure in a Global Environment, but all errors are ours alone

the relevant ~ i ia rke t 13rice.s 1x11 :llso on the ~ ~ o s i t i o n vector rh:lt clescril)es the I " A moment's reflection reveals inissoeci17cation to be Ihe comoellino case. The ~ n o d - I,orlfolio. E.stiln;l[es h-cl;lv ~ o l : ~ t i [ i l ~ ;Ire ;,reclicatecl o n :~s,sullil,tion ol. I em approach is to acknowledge inisspecification from the o ise t , as lor example in , * ;I I'iseeI position vector t~ir;I~ghout the h-cI;ly horizon, which is nn i l te l y . lheinfluentialpaperof

'%oreover, Cl~ristoffersen & Diebold (1997) show that the predictable volatility dy- I'ositions tend to cIi:inge frecl~lently i n rlie course 01' no~,n i :~ l tlxcling, namics in manv returns vanish ouicl~lv with horizon, ,ndicatino sealiflo - - - - , ---- - - - - - ~ , - ~ , - -~ . ~ ~- ., - - ~ - -

I,otIi wi th in ;~ncl cross clays, for ;I n l l m l ~ e r o f re;isons. Pi~..st, 11ositions m:Iy I leadone astray Ipe t:llien to filcilitate :I customer Ifi~nsaction, ;~ncl then tlecline to normal 13 See ~ j n d l ~ ~ (7983) and Diebold (7998) fordjscussjon ofthis same point in /he con- inue i i to i y " levels w l i e ~ i o i ke l t i ng custoliier o ~ l e l r o ) m e in, o r ivhen t i le text of inore fraditIonalforecastin~probIe~ns

-- - . REFERENCES

Bollerslev T, 1986 in Macroeconomelrics: developments, tensions and prospecls. Kevin Hoover, edilor, Generalized autoregressive conditional heteroskedaslicily Kluwer Academic Press, Boston. pages 427-472 Journal of Econometrics 31, pages 307-327 Drost F and T Nijman, 1993 Bollerslev T, R Chou and K Kroner, 1992 Temporal aggregal~on of Garch processes Arch modeling ~n linance: a seleclive review 01 the lheory and empirical evidence Economelr~ca 61, pages 909-927 Journal of Econometrics 52, pages 5-59 Falloon W, 1995 Campbell J, A Lo and A MacKinlay, 1997 2020 Visions The econom~cs of financial markels Risk October, pages 43-45 Princeton: Princeton Unlversily Press Findley D, 1983 Chew L, 1994 On the use of multiple models for multi-period forecasting Shock treatment American Statislical Association. Proceedings of Ihe Buslness and Econom~c Slal~sltc Risk September, pages 63-70 Sectlon, pages 528-531

Christoffersen P and F Diebold, 1997 JP Morgan, 1996 How relevant IS volal~lily forecasting for financial risk management? RiskMelncs - lechnical document Wharton Financial Institutions Cenler-worklng paper 97-45, Unlversity of Fourlh edition, New York Pennsylvania Kupiec P and J O'Brien, 1995 Cootner P, 1964 Internal affa~rs The random character of stock markel prices Risk May, pages 43-47 MIT Press. Cambridge Nelson D and D Foster, 1994 Diebold F, 1988 Asymptotic filtering theory for univanate Arch models Emp~ncal modeling of exchange rate dynamics Econometrica 62, pages 1-41 Sprtnger-Verlag. New York Smithson C and L Minton, 1996a Diebold F, 1998 Value-at-risk Nemencs of forecasting in business, economics, government and finance Risk January, pages 25-27 South-'Western College Pubiish~ng. Cinc~nnali, Ohio Srnithson C and L Minton, 1996b Diebold F and J Lopez, 1995 l/alue-at-risk (2) Model~ng volatility dynamics Risk February, pages 38-39

107 RISK JANUARY 1998